Proper Almost-Homogeneous Domains of the Einstein Universe

TL;DR

This paper classifies a unique almost-homogeneous bounded domain in the Einstein universe, called a diamond, which models certain symmetric spaces and aids in understanding conformally flat manifolds.

Contribution

It identifies and classifies the only almost-homogeneous bounded domain in Einstein universes, linking it to symmetric spaces and conformally flat manifold classifications.

Findings

Existence of a unique bounded almost-homogeneous domain called a diamond.

The diamond models the symmetric space of PO(p,1) × PO(1,q).

Classification of closed conformally flat manifolds with proper development.

Abstract

The Einstein universe $\mathbf{Ein}^{p,q}$ of signature $(p,q)$ is a pseudo-Riemannian analogue of the conformal sphere; it is the conformal compactification of the pseudo-Riemannian Minkowski space. For $p,q \geq 1$, we show that, up to a conformal transformation, there is only one almost-homogeneous domain in $\mathbf{Ein}^{p,q}$ that is bounded in a suitable stereographic projection. This domain, which we call a diamond, is a model for the symmetric space of $\operatorname{PO}(p,1) \times \operatorname{PO}(1,q)$. We deduce a classification of closed conformally flat manifolds with proper development.